C₂²×Q₈×C₁₀₁ #15692156

order	3232
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×Q₈×C₁₀₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₁	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₁₀₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ G

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 4 generators on 113 points:

(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)

Conjugacy classes & character table

2020 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#391×C₁₀₁, #393×C₁₀₁, #395×C₁₀₁, …
maximal normal subgroup of	#391×C₁₀₁, #393×C₁₀₁, #395×C₁₀₁, …
maximal quotient of	C₂²×C₄⋊C₄×C₁₀₁, C₂×C₄×Q₈×C₁₀₁, C₂×C₂²⋊Q₈×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3232, 47);

C22×Q8×C101 #15692156